I have the same problems with my LG Gossip. When the headset is plugged in, it will shut off during conversations at least every 5-10 minutes. I had the battery problem happen to me once in terms of it saying the battery is full and it in fact being ...
I believe that the LG headset that came with the phone may also cause the phone to shut off. I was using it today during a phone call and the phone shut off 4 times. When I disconnected it and talked directly into the phone, it did not shut off.